Calculus III, Fall 2007

Welcome to this course page! This is a freely available page about the course. However, if you have enrolled in the class, please go to Courseworks to find your homework assignments. You are responsible for doing this, whether or not I remind you in class!

Course information

Assessment

Homework (30%)
Two midterm exams (20% each)
Final exam (30%)

The homework will mostly be exercises from Stewart, though a few of them may be my own questions. A sample of the questions will be graded. There will be 9 assignments, but your lowest score will not count towards your overall grade.

Questions and comments

Feedback and questions are welcome. If you have questions or comments, I'd like to talk to you in person, during my office hours or in or directly after class. (If you're worried about your grades, asking questions about the course before you hand in your assignment is a much better bet than asking for special consideration afterwards.) You can also email me if you wish.

Honesty

Cheating in exams is of course unacceptable. Any cheating during midterms or finals will result in you failing the course and the matter being reported to your dean.

Copying your written work from somebody else or from any other source is considered cheating and will be dealt with severely. Permitting someone else to copy your work is also considered cheating.

You can talk to your friends about the homework, but remember that the work you hand in must be your own. To avoid any suspicion of copying, I strongly recommend that you write out your homework solutions whilst working on your own, not whilst working with a friend.

Using the web

Stewart's webpage contains various supplements to the book which you may find useful.

Some general comments about using the internet in your courses. You must not use web resources dishonestly. However, you can use web searches constructively as an extra resource. If you're stuck because you don't have a calculator to hand, google one. If you're stuck because your notes refer to something called an icosahedron, and you don't what an icosahedron is, google will rapidly find you informative articles in Wikipedia and elsewhere. These may contain mistakes and should not be quoted as authorities (because they are anonymous and unreliable) but are often useful all the same.

About the course

The first half of this course is not about calculus but about algebraic techniques for describing geometry in two and three dimensions. For example, there are special ways of "multiplying" points in the plane (the product of complex numbers) and points in space (the cross product of vectors), both of which have geometric as well as algebraic significance.

The second half of the course is about differential calculus applied to functions of two variables and to curves in space. The lines and planes studied in the first part here appear as first order approximations, i.e. derivatives. Conics (ellipses and hyperbolas) appear as second order approximations, allowing us to distinguish maxima, minima and saddle points.

In physics and engineering, complex numbers are used to describe waves, notably the wavefunctions of quantum mechanics; vector algebra is used to describe (among many other things) electric and magnetic fields. Space curves describe the trajectories of objects moving in those fields. In economics, it is often useful to be able to maximise functions, especially when there are additional constraints.

Course text

J. Stewart, Calculus: Early Transcendentals, 6th edition (new!), Brooks-Cole. Available in the university bookstore.

The aim, essentially, is to cover Chapters 12, 13, 14 (plus a few other bits and pieces). However, I will not stick rigidly to the order or content of these chapters.

Syllabus (tentative)

I have marked with an asterisk those (few) topics not covered by Stewart.

Part 1.	Algebra and geometry in the plane and in space.
Sept. 5	Introduction: Linear and quadratic functions in two variables. Idea of linear or quadratic approximations to f(x,y).	10.5
Sept. 10	Coordinate systems in 2 and 3 dimensions (rectangular, polar, cylindrical, spherical). Coordinate change formulas.	10.3, 12.1, 15.7 (first bit), 15.8 (first bit)
Sept. 12	Vectors. Addition and scalar multiplication. Length.	12.2
Sept. 17	Dot product.	12.3
Sept. 19	More about the dot product. Components. Change of coordinate. Reflections.	12.3
Sept. 24	Cross product. *Rotation invariance. Sine formula.	12.4
Sept. 26	Equations of lines and planes in three dimensions.	12.5
Oct. 1	Review.
Oct. 3	FIRST MIDTERM.
Oct. 8	Complex numbers. Multiplication. Modulus and conjugation. Division. Quadratics.	App. H
Oct. 10	Complex exponentials. Geometric interpretation. Roots of unity.	App. H
Oct. 15	Quadric surfaces.	12.6
Part 2.	Differentiating functions of several variables.
Oct. 17	Functions of several variables. Continuity.	14.1, 14.2
Oct. 22	Partial Derivatives. Equality of cross partials. PDE from physics.*	14.3
Oct. 24	Tangent planes and linear approximations.	14.4
Oct. 29	Chain rule. Laplacian in polar, cylindrical and spherical coordinates	14.5
Oct. 31	Directional derivatives, tangent plane.	14.6
Nov. 5	Academic holiday.
Nov. 7	Maxima and minima.	14.7
Nov. 12	Second order behavior near a critical point, saddle points.	14.7
Nov. 14	Review.
Nov. 19	SECOND MIDTERM.
Part 3.	Further topics
Nov. 21	Complex dynamics. The Mandelbrot set.
Nov. 26	Lagrange multipliers.	14.8
Nov. 28	Space curves.	13.1, 13.2
Dec. 3	Curvature and torsion. Frenet formulas.	13.3
Dec. 5	Motion in space. Newton's law; Kepler's solution.	13.4
Dec. 10	Review.

Homework schedule

Hand in by noon Thursday.

Please go to Courseworks to find the assignments themselves.

Assignment 1: Thurs. Sept 13
Assignment 2: Thurs. Sept 20
Assignment 3: Thurs. Sept. 27
Assignment 4: Thurs. Oct. 11
Assignment 5: Thurs. Oct. 18
Assignment 6: Thurs. Oct. 25
Assignment 7: Thurs. Nov. 15
Assignment 8: Thurs. Nov. 29
Assignment 9: Thurs. Dec. 6